EP2385213A1 - System und Verfahren zur Modellierung von Bohrlochverläufen - Google Patents

System und Verfahren zur Modellierung von Bohrlochverläufen Download PDF

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Publication number
EP2385213A1
EP2385213A1 EP11173354A EP11173354A EP2385213A1 EP 2385213 A1 EP2385213 A1 EP 2385213A1 EP 11173354 A EP11173354 A EP 11173354A EP 11173354 A EP11173354 A EP 11173354A EP 2385213 A1 EP2385213 A1 EP 2385213A1
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Prior art keywords
survey point
vector
wellbore
survey
model
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Granted
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EP11173354A
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English (en)
French (fr)
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EP2385213B1 (de
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Robert Mitchell
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Landmark Graphics Corp
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Landmark Graphics Corp
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    • E—FIXED CONSTRUCTIONS
    • E21—EARTH OR ROCK DRILLING; MINING
    • E21B—EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B47/00—Survey of boreholes or wells
    • E21B47/02—Determining slope or direction
    • E21B47/022—Determining slope or direction of the borehole, e.g. using geomagnetism

Definitions

  • the present invention generally relates to modeling wellbore trajectories. More particularly, the present invention relates to the use of spline functions, derived from drill string solutions, to model wellbore trajectories.
  • Wellbore trajectory models are used for two distinct purposes.
  • the first use is planning the well location, which consists of determining kick-off points, build and drop rates, and straight sections needed to reach a specified target.
  • the second use is to integrate measured inclination and azimuth angles to determine a well's location.
  • the tangential model consists of straight line sections.
  • the slope of this model is discontinuous at survey points.
  • the most commonly used model is the minimum curvature model, which consists of circular-arc sections. This model has continuous slope, but discontinuous curvature.In fact, the minimum curvature model argues that a wellbore would not necessarily have continuous curvature.
  • Torque-drag modeling refers to the torque and drag related to drillstring operation. Drag is the excess load compared to rotating drillstring weight, which may be either positive when pulling the drillstring or negative while sliding into the well. This drag force is attributed to friction generated by drillstring contact with the wellbore. When rotating, this same friction will reduce the surface torque transmitted to the bit. Being able to estimate the friction forces is useful when planning a well or analysis afterwards. Because of the simplicity and general availability of the torque-drag model, it has been used extensively for planning and in the field. Field experience indicates that this model generally gives good results for many wells, but sometimes performs poorly.
  • the drillstring trajectory is assumed to be the same as the wellbore trajectory, which is a reasonable assumption considering that surveys are taken within the drillstring.
  • Contact with the wellbore is assumed to be continuous.
  • the most common method for determining the wellbore trajectory is the minimum curvature method, the wellbore shape is less than ideal because the bending moment is not continuous and smooth at survey points. This problem is dealt with by neglecting bending moment but, as a result of this assumption, some of the contact force is also neglected.
  • the present invention meets the above needs and overcomes one or more deficiencies in the prior art by providing systems and methods for modeling a wellbore trajectory, which can be used to model the corresponding drillstring trajectory and transform the torque-drag drill string model into a full stiff-string formulation.
  • the present invention includes a computer implemented method for modeling a wellbore trajectory, which comprises: i) calculating a tangent vector interpolation function for each interval between two or more survey points within a wellbore using a wellbore curvature, a tangent vector and a normal vector at each respective survey point; and (ii) determining the wellbore trajectory using each tangent vector interpolation function in a torque-drag drillstring model.
  • the method may further comprise calculating the tangent vector at each survey point using survey data at each respective survey point.
  • the survey data may comprise an inclination angle, an azimuth angle and a measured depth at each survey point.
  • the method may further comprise calculating the wellbore curvature at each survey point using a first derivative of the tangent vector, a coefficient and another coefficient at each respective survey point; and calculating the normal vector at each survey point using the first derivative of the tangent vector, the coefficient and the another coefficient at each respective survey point.
  • the first derivative of the tangent vector may be continuous at each survey point.
  • the method may further comprise calculating the coefficient at each survey point in a direction of a special normal vector at the respective survey point using a block tridiagonal matrix; and calculating the another coefficient at each survey point in a direction of a special binormal vector at the respective survey point using the block tridiagonal matrix.
  • the method may further comprise calculating the block tridiagonal matrix using the tangent vector, the special normal vector, and the special binormal vector at each respective survey point.
  • the method may further comprise calculating the special normal vector at each survey point; and calculating the special binormal vector at each survey point.
  • the method may further comprise calculating a torque-drag drilistring solution using the wellbore trajectory.
  • the method may further comprise refining the wellbore trajectory using the torque-drag drillstring solution.
  • the present invention includes a computer readable medium having computer executable instructions for modeling a wellbore trajectory.
  • the instructions are executable to implement: i) calculating a tangent vector interpolation function for each interval between two or more survey points within a wellbore using a wellbore curvature, a tangent vector and a normal vector at each respective survey point; and (ii) determining the wellbore trajectory using each tangent vector interpolation function in a torque-drag drillstring model.
  • the present invention includes a computer implemented method for modeling a wellbore trajectory, which comprises: i) calculating a tangent vector at each survey point within a wellbore using survey data at each respective survey point, the wellbore comprising two or more survey points; ii) calculating a special normal vector and a special binormal vector at each survey point; iii) calculating a block tridiagonal matrix using the tangent vector, the special normal vector, and the special binormal vector at each respective survey point; iv) calculating a coefficient at each survey point in the direction of the special normal vector at the respective survey point and another coefficient at each survey point in the direction of the special binormal vector at the respective survey point using the block tridiagonal matrix; v) calculating a wellbore curvature at each survey point and a normal vector at each survey point using a first derivative of the tangent vector, the coefficient and the another coefficient at each respective survey point; vi) calculating a tangent vector interpolation function for each interval between the survey points
  • the present invention includes a computer readable medium having computer executable instructions for modeling a wellbore trajectory, The instructions are executable to implement: i) calculating a tangent vector at each survey point within a wellbore using survey data at each respective survey point, the wellbore comprising two or more survey points; ii) calculating a special normal vector and a special binormal vector at each survey point; iii) calculating a block tridiagonal matrix using the tangent vector, the special normal vector, and the special binormal vector at each respective survey point; iv) calculating a coefficient at each survey point in the direction of the special normal vector at the respective survey point and another coefficient at each survey point in the direction of the special binormal vector at the respective survey point using the block tridiagonal matrix; v) calculating a wellbore curvature at each survey point and a normal vector at each survey point using a first derivative of the tangent vector, the coefficient and the another coefficient at each respective survey point; vi) calculating a tangent vector
  • FIG. 1 is a block diagram illustrating one embodiments of a system for implementing the present invention.
  • FIG. 2 is a graphical illustration comparing the analytic model, the minimum curvature model and the spline model of the present invention for a circular-arc wellbore trajectory.
  • FIG. 3 is a graphical illustration comparing the analytic model, the minimum curvature model and the spline model of the present invention for a catenary wellbore trajectory.
  • FIG. 4 is a graphical illustration comparing the analytic model, the minimum curvature model and the spline model of the present invention for a helix wellbore trajectory.
  • FIG. 5 is a graphical illustration comparing the rate-of-change of curvature between an analytic model and the spline model of the present invention for a catenary wellbore trajectory.
  • FIG. 6 is a graphical illustration comparing the torsion between an analytic model and the spline model of the present invention for a helix wellbore trajectory.
  • FIG. 7 illustrates the test case wellbore used in Example 1.
  • FIG. 8 is a graphical illustration comparing the bending moment between the minimum curvature model and the spline model of the present invention for the test case wellbore used in Example 1.
  • FIG. 9A is a graphical illustration (vertical view) of the short radius wellpath used in
  • FIG. 9B is a graphical illustration (North/East view) of the short radius wellpath used in Example 2.
  • FIG. 10 is a graphical illustration comparing the short radius contact force between a constant curvature model and the spline model of the present invention for the wellpath used in Example 2.
  • FIG. 11 is a graphical illustration comparing the short radius bending moment between a constant curvature model and the spline model of the present invention for the wellpath used in Example 2.
  • FIG. 12 is a flow diagram illustrating one embodiment of a method for implementing the present invention.
  • the present invention may be implemented through a computer-executable program of instructions, such as program modules, generally referred to as software applications or application programs executed by a computer.
  • the software may include, for example, routines, programs, objects, components, and data structures that perform particular tasks or implement particular abstract data types.
  • the software forms an interface to allow a computer to react according to a source of input.
  • WELLPLANTM which is a commercial software application marketed by Landmark Graphics Corporation, may be used as an interface application to implement the present invention.
  • the software may also cooperate with other code segments to initiate a variety of tasks in response to data received in conjunction with the source of the received data.
  • the software may be stored and/or carried on any variety of memory media such as CD-ROM, magnetic disk, bubble memory and semiconductor memory (e.g ., various types of RAM or ROM). Furthermore, the software and its results may be transmitted over a variety of carrier media such as optical fiber, metallic wire, free space and/or through any of a variety of networks such as the Internet.
  • memory media such as CD-ROM, magnetic disk, bubble memory and semiconductor memory (e.g ., various types of RAM or ROM).
  • the software and its results may be transmitted over a variety of carrier media such as optical fiber, metallic wire, free space and/or through any of a variety of networks such as the Internet.
  • the invention may be practiced with a variety of computer-system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable-consumer electronics, minicomputers, mainframe computers, and the like. Any number of computer-systems and computer networks are acceptable for use with the present invention,
  • the invention may be practiced in distributed-computing environments where tasks are performed by remote-processing devices that are linked through a communications network.
  • program modules may be located in both local and remote computer-storage media including memory storage devices.
  • the present invention may therefore, be implemented in connection with various hardware, software or a combination thereof, in a computer system or other processing system.
  • FIG. 1 a block diagram of a system for implementing the present invention on a computer is illustrated.
  • the system includes a computing unit, sometimes referred to as a computing system, which contains memory, application programs, a client interface, and a processing unit.
  • the computing unit is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention.
  • the memory primarily stores the application programs, which may also be described as program modules containing computer-executable instructions, executed by the computing unit for implementing the methods described herein and illustrated in FIGS. 2-12 .
  • the memory therefore, includes a Wellbore Trajectory Module, which enables the methods illustrated and described in reference to FIGS 2-12 , and WELLPLAN TM .
  • the computing unit typically includes a variety of computer readable media.
  • computer readable media may comprise computer storage media and communication media.
  • the computing system memory may include computer storage media in the form of volatile and/or nonvolatile memory such as a read only memory (ROM) and random access memory (RAM).
  • ROM read only memory
  • RAM random access memory
  • a basic input/output system (BIOS) containing the basic routines that help to transfer information between elements within the computing unit, such as during start-up, is typically stored in ROM.
  • the RAM typically contains data and/or program modules that are immediately accessible to, and/or presently being operated on by, the processing unit.
  • the computing unit includes an operating system, application programs, other program modules, and program data.
  • the components shown in the memory may also be included in other removable/nonremovable, volatile/nonvolatile computer storage media.
  • a hard disk drive may read from or write to nonremovable, nonvolatile magnetic media
  • a magnetic disk drive may read from or write to a removable, non-volatile magnetic disk
  • an optical disk drive may read from or write to a removable, nonvolatile optical disk such as a CD ROM or other optical media.
  • Other removable/non-removable, volatile/non-volatile computer storage media that can be used in the exemplary operating environment may include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like.
  • the drives and their associated computer storage media discussed above therefore, store and/or carry computer readable instructions, data structures, program modules and other data for the computing unit.
  • a client may enter commands and information into the computing unit through the client interface, which may be input devices such as a keyboard and pointing device, commonly referred to as a mouse, trackball or touch pad.
  • input devices may include a microphone, joystick, satellite dish, scanner, or the like.
  • a monitor or other type of display device may be connected to the system bus via an interface, such as a video interface.
  • computers may also include other peripheral output devices such as speakers and printer, which may be connected through an output peripheral interface.
  • the present invention proceeds from the concept that the trajectory given by the survey measurements made within the drillstring is the trajectory of the drill string, which must have continuity of bending moment proportional to curvature.
  • the nomenclature used herein is described in the Society of Petroleum Engineers article "Drillstring Solutions Improve the Torque-Drag Model” by Mitchell, Robert F. ("SPE 112623”), which is incorporated herein by reference and repeated in Table 1 below.
  • the f j are cubic functions of x and the unknown coefficients y j n are determined by requiring continuity of the first derivative of y(x) at each x j .
  • the functions in equation (2) need not be cubic functions. They must only satisfy equations (3).
  • the use of spline formulations such as, for example, cubic splines and tangent splines to model wellbore trajectories is well known in the art. The determination of the wellbore trajectory from survey data, however is not.
  • the use of conventional splines, as applied to a three-dimensional curve will not satisfy equation (5) and equations (6).
  • s is measured depth
  • ⁇ j is the curvature at s j
  • n j is the normal vector at s j .
  • This formulation has two purposes.
  • the second reason is to insure that s is indeed measured depth.
  • equation (4) satisfies this condition.
  • the normal method for determining the well path is to use some type of surveying instrument to measure the inclination and azimuth at various depths and then to calculate the trajectory.
  • Each survey point j therefore, includes survey data comprising an inclination angle ⁇ j , an azimuth angle ⁇ j and a measured depth s.
  • These angles have been corrected (i) to true north for a magnetic survey or (ii) for drift if a gyroscopic survey.
  • the survey angles define the tangent i j to the trajectory at each survey point j where the tangent vector is defined in terms of inclination ⁇ j and azimuth ⁇ j in the following formulas:
  • a tangent vector interpolation function connecting these vectors can be created.
  • the function T j satisfies the Frenet equations d ⁇ t
  • t j , ⁇ j ,and b j form a right-handed coordinate system at s j .
  • the variables ⁇ j - and ⁇ j are not the most convonient choices because of the nonlinearity introduced by the sine and cosine functions, An
  • Equation (30a) can be used to define what are known as tension-splines and equation (30b) may be used to define "compression" splines. This is demonstrated in the following section using drillstring solutions as interpolation functions.
  • ⁇ coefficients are functions of the axial force, which are not known until the torque-drag equations are solved.
  • ⁇ tends to be small, so that the solution approximates a cubic equation.
  • the cubic interpolation can be used to approximate the trajectory, and to solve the torque-drag problem.
  • the torque-drag solution can then be used to refine the trajectory, iterating if necessary.
  • FIGS. 2-4 A simple comparison of the wellbore trajectory model of the present invention, also referred to as a spline model, and the standard minimum curvature model with three analytic wellbore trajectories (circular-arc catenary, helix) is illustrated in FIGS. 2-4 , respectively.
  • the comparisons of the displacements illustrated in FIGS. 2-4 demonstrate that the minimum curvature model and the spline model match the analytic wellbore trajectory in FIG. 2 (circular-arc), the analytic wellbore trajectory in FIG. 3 (catenary) and the analytic wellbore trajectory in FIG. 4 (helix), Only one displacement is shown for the helix, but is representative of the other displacements.
  • the spline model was also used to calculate the rate of change of curvature for the analytic wellbore trajectory illustrated in FIG. 5 (catenary), and the geometric torsion for the analytic wellbore trajectory illustrated in FIG. 6 (catenary), and the geometric torsion for the analytic wellbore trajectory illustrated in FIG. 6 (helix).
  • the results illustrated by the comparisons in FIGS. 5-6 demonstrate the deficiencies of the minimum curvature model when calculating the curvature rate of change for the catenary wellbore trajectory illustrated in FIG. 5 or when calculating the geometric torsion for the helix wellbore trajectory illustrated in FIG. 6 .
  • the minimum curvature model predicts zero for both quantities compared in FIGS. 5-6 , which cannot be plotted.
  • the spline model determines both quantities accurately, although there is some end effect apparent in the geometric torsion calculation. Additional advantages attributed to the present invention (spline model) are demonstrated by the following examples.
  • Torque-drag calculations were made using a comprehensive torque-drag model well known in the art. Similarly, the equilibrium equations were integrated using a method well known in the art. Otherwise, the only difference in the solutions is the choice of the trajectory model.
  • the drag and torque properties of an idealized well plan are based on Well 3 described in Society of Petroleum Engineers article "Designing Well Paths to Reduce Drag and Torque" by Sheppard, M.C., Wick, C. and Burgess, T.M.
  • the fixed points on the model trajectory are as follows: i) the well is considered to be drilled vertically to a KOP at a depth of 2,400 ft.; ii) the inclination angle then builds at a rate of 5°/100 ft; and iii) the target location is considered to be at a vertical depth of 9,000 ft and displaced horizontally from the rig location by 6,000 ft.
  • the model drillstring was configured with 372 feet of 6-112 inch drill collar (99.55 Ibf./ft.) and 840 ft of 5 inch heavyweight pipe (50.53 lbf./ft.) with 5 inch drillpipe (20.5 lbf./ft.) to the surface.
  • a mud weight of 9.8 lbm/gal was used.
  • a value of 0.4 was chosen for the coefficient of friction to simulate severe conditions. Torque-loss calculations were made with an assumed WOB of 38,000 lbf. and with an assumed surface torque of 24,500 ft.-Ibf
  • Hook load calculated for zero friction was 192202 lbf. for the circular-arc calculation, and 192T64 lbf. for the spline model, which compare to a spreadsheet calculation of 192203 lbf.
  • the slight difference (38 lbf.) is due to the spline taking on a slightly different shape (due to smoothness requirements) from the straight-line/circular-arc shapes specified, which the minimum curvature model exactly duplicated.
  • all other aspects of the axial force calculations are identical between the two models. Tripping out, with a friction coefficient of 0.4, the hook load was 313474 lbf. for the circular-arc model and 319633 lbf.
  • FIGS. 9A and 9B the vertical and horizontal views of the end of the wellpath are illustrated, respectively.
  • the build rate for this example was 42°/30m, roughly ten times the build rate of the first case in Example 1.
  • FIG. 10 some of the contact force is neglected by neglecting the bending moment since the contact force for the spline model at the end of the build is four times that of the minimum curvature model.
  • FIG. 11 the bending moment for this example is illustrated.
  • the minimum curvature model still provides a lower bending moment than the spline model, but the spline results are still much smoother.
  • FIG. 12 flow diagram illustrates one embodiment of a method 1200 for implementing the present invention.
  • step 1202 the survey data is obtained for each survey point ( j ).
  • a tangent vector ( t j ) is calculated at each survey point using the survey data at each respective survey point.
  • step 1206 a special normal vector ( ⁇ j ) and a special binormal vector ( b ⁇ j ) are calculated at each survey point.
  • a block tridiagonal matrix is calculated using the tangent vector the special normal vector and the special binormal vector at each respective survey point.
  • a coefficient ( ⁇ j ) is calculated at each survey point in the direction of the special normal vector at the respective survey point and another coefficient ( ⁇ j ) is calculated at each survey point in the direction, of the special binormal vector at the respective survey point using the block tridiagonal Matrix.
  • a wellbore curvature ( ⁇ j ) and a normal vector ( n j ) are calculated at each survey point using a first derivative of the tangent vector, the coefficient and the another coefficient at each respective survey point.
  • a tangent vector interpolation function ( t j ( s )) is calculated for each interval between survey points using the wellbore curvature, the tangent vector and the normal vector at each respective survey point.
  • step 1216 the wellbore trajectory is determined using each tangent vector interpolation function in a torque-drag drillstring model.

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EP11173354A 2007-12-17 2008-12-16 System und Verfahren zur Modellierung von Bohrlochverläufen Not-in-force EP2385213B1 (de)

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M.C SHEPPARD ET AL: "Designing Well Paths To Reduce Drag and Torque", SPE DRILLING EMGINEERING (SPE 15463), December 1987 (1987-12-01), pages 344 - 350, XP002536071 *
ROBERT F. MITCHELL ET AL: "How Good is the Torque-Drag Model?", SPE/IADC 105068, 20 February 2007 (2007-02-20), pages 1 - 9, XP002536069 *

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EP2232012B1 (de) 2011-10-19
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US8160853B2 (en) 2012-04-17
US20090157319A1 (en) 2009-06-18
AU2008338485B2 (en) 2013-02-14
ATE529608T1 (de) 2011-11-15
EP2385213B1 (de) 2013-03-06
EP2232012A2 (de) 2010-09-29
CN101983276A (zh) 2011-03-02
US20120179445A1 (en) 2012-07-12
MX2010006749A (es) 2010-09-30
US8594987B2 (en) 2013-11-26
AU2008338485A1 (en) 2009-06-25
WO2009079492A3 (en) 2009-10-01

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